A Note on Steiner Tree Games Darko Skorin-Kapov School of Business, Adelphi University, Garden City, New York 11530 Jadranka Skorin-Kapov College of Business, State University of New York at Stony Brook, Stony Brook, New York 11794 We investigate the cost allocation strategy associated with the problem of providing some service of com- mon interest from some source to a number of network users, via the minimum cost directed Steiner tree (ST) network that spans the source and all the receivers. The cost of such ST is distributed among its receivers who may be individuals or organizations with possi- bly conflicting interests. The objective of this article is to develop a reasonably fair and efficient cost alloca- tion scheme associated with the above cost allocation problem. Since finding the optimal Steiner tree is an NP-hard problem, the input to our cost allocation prob- lem is the best known Steiner tree obtained using some heuristic. To allocate the cost of this Steiner tree to the users (receiver nodes), we formulate the associated Modified Steiner Tree Network (MSTN) game in charac- teristic function form. Then we construct a primal-dual cost allocation algorithm which finds some points in the core of the MSTN game and thus results in subsidy-free cost allocations. Moreover, for the given Steiner tree, our cost allocation scheme is efficient and finds the above “fair” cost allocations in polynomial time. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 59(2), 215–225 2012 Keywords: communication network; cost allocation; coopera- tive games; mathematical programming; Steiner trees 1. INTRODUCTION Let us consider a connected directed network. Assume that a common source provides a service, which is required by users residing at some network nodes, and assume that any node receiving the service can in turn deliver it to its adja- cent nodes. Each user is required to be connected, perhaps through other (switching) nodes, to the common source. With each edge (link) we associate the cost of using that edge to provide service. The set of users should be linked to the com- mon source at a minimum cost. We refer to this problem as Received December 2007; accepted January 2011 Correspondence to: D. Skorin-Kapov; e-mail: skorin@adelphi.edu DOI 10.1002/net.20436 Published online 17 March 2011 in Wiley Online Library (wileyonlinelibrary.com). © 2011 Wiley Periodicals, Inc. to the Min Cost Steiner Tree problem in networks, or simply as the Steiner tree (ST) problem. It is well known that finding an optimal Steiner tree in networks is a computationally pro- hibitive problem and numerous researchers and practitioners have proposed various exact and heuristic approaches to solve it. (For a set of related articles and a survey see respectively [3] and [33]). In our study, the computational complexity of the ST prob- lem is further complicated by cost allocation considerations. The cost of a service network is shared by users who possibly have conflicting objectives. However, they might cooperate to decrease their joint cost. These individuals or organizations are likely to support globally “attractive” solutions only if their expectations for a “fair share” of the cost are met. There is no general answer to the fairness issue for net- work cost allocation problems. Cooperative game theory has been used to analyze several classes of such problems in the literature. Some examples include: spanning tree games [2], Steiner tree games [27], network flow games [9], cost allocation arising from routing in networks [21], capaci- tated network design games [25, 26] and hub network games [29,30]. For a survey and numerous references on cost alloca- tion models in networks see for example [18, 24]. A common approach in the above papers is to formulate the associated cost allocation problem as a cooperative game in character- istic function form, followed by the evaluation of various game theoretic solution concepts in the context of a particu- lar problem. We take a similar approach in this study of the cost allocation problem in Steiner trees. Cooperative game theory offers the concept of a “fair” cost allocation solution known as the core of a cooperative game. The core consists of so-called stable cost allocation solutions which provide no incentive for any coalition of users to secede and build their own subnetwork, i.e., it avoids cross-subsidies. Let us first informally describe the related Steiner Tree Network (STN) game, previously considered in the literature. Given a complete network G, the value of the charac- teristic function for each subset of user nodes is defined as the optimal cost of the Steiner tree in G which spans all the users of that subset. Researchers have extensively NETWORKS—2012—DOI 10.1002/net